Related papers: Deformed single ring theorems

We consider the random matrix model $X_n = P_n + i Q_n$, where $P_n$ and $Q_n$ are independently Haar-unitary rotated Hermitian matrices with at most $2$ atoms in their spectra. Let $(M, \tau)$ be a tracial von Neumann algebra and let $p, q…

Consider a $n\times n$ sparse non-Hermitian random matrix $X_n$ defined as the Hadamard product between a random matrix with centered independent and identically distributed entries and a sparse Bernoulli matrix with success probability…

Consider the random matrix $\Sigma = D^{1/2} X \widetilde D^{1/2}$ where $D$ and $\widetilde D$ are deterministic Hermitian nonnegative matrices with respective dimensions $N \times N$ and $n \times n$, and where $X$ is a random matrix with…

We show that a perturbation of any fixed square matrix D by a random unitary matrix is well invertible with high probability. A similar result holds for perturbations by random orthogonal matrices; the only notable exception is when D is…

We investigate eigenvectors of rank-one deformations of random matrices $\boldsymbol B = \boldsymbol A + \theta \boldsymbol {uu}^*$ in which $\boldsymbol A \in \mathbb R^{N \times N}$ is a Wigner real symmetric random matrix, $\theta \in…

We consider large Hermitian matrices whose entries are defined by evaluating the exponential function along orbits of the skew-shift $\binom{j}{2} \omega+jy+x \mod 1$ for irrational $\omega$. We prove that the eigenvalue distribution of…

We consider the local eigenvalue distribution of large self-adjoint $N\times N$ random matrices $\mathbf{H}=\mathbf{H}^*$ with centered independent entries. In contrast to previous works the matrix of variances $s_{ij} = \mathbb{E}\,…

Horn's problem -- to find the support of the spectrum of eigenvalues of the sum $C=A+B$ of two $n$ by $n$ Hermitian matrices whose eigenvalues are known -- has been solved by Knutson and Tao. Here the probability distribution function (PDF)…

We consider random hermitian matrices in which distant above-diagonal entries are independent but nearby entries may be correlated. We find the limit of the empirical distribution of eigenvalues by combinatorial methods. We also prove that…

For correlated real symmetric or complex Hermitian random matrices, we prove that the local eigenvalue statistics at any cusp singularity are universal. Since the density of states typically exhibits only square root edge or cubic root cusp…

We derive the joint asymptotic distribution of the outlier eigenvalues of an additively deformed Wigner matrix $H$. Our only assumptions on the deformation are that its rank be fixed and its norm bounded. Our results extend those of [The…

We consider the Anderson model on the finite grid $G = \mathbb Z/L_1\mathbb Z\times\cdots\times\mathbb Z/L_d\mathbb Z$, defined by the random Hamiltonian $H_t=\Delta+tV$, where $\Delta$ is the discrete Laplacian and…

We study $N \times N$ random band matrices $H = (H_{xy})$ with mean-zero complex Gaussian entries, where $x,y$ lie on the discrete torus $(\mathbb{Z} / \sqrt[d]{N} \mathbb{Z})^d$ in dimensions $d \ge 3$. The variance profile satisfies…

We consider the distribution of the top eigenvector $\widehat{v}$ of a spiked matrix model of the form $H = \theta vv^* + W$, in the supercritical regime where $H$ has an outlier eigenvalue of comparable magnitude to $\|W\|$. We show that,…

We study a particular class of families of multi-dimensional lattice Schr\"o\-dinger operators with deterministic (including quasi-periodic) potentials generated by the "hull" given by an orthogonal series over the Haar wavelet basis on the…

Universality of local eigenvalue statistics is one of the most striking phenomena of Random Matrix Theory, that also accounts for a lot of the attention that the field has attracted over the past 15 years. In this paper we focus on the…

This note deals with a problem of the probabilistic Ramsey theory in functional analysis. Given a linear operator $T$ on a Hilbert space with an orthogonal basis, we define the isomorphic structure $\Sigma(T)$ as the family of all subsets…

Given an $n\times n$ random matrix $X_n$ with i.i.d. entries of unit variance, the circular law says that the empirical spectral distribution (ESD) of $X_n/\sqrt{n}$ converges to the uniform measure on the unit disk. Let $M_n$ be a…

We study n by n symmetric random matrices H, possibly discrete, with iid above-diagonal entries. We show that H is singular with probability at most exp(-n^c), and the spectral norm of the inverse of H is O(sqrt{n}). Furthermore, the…

Consider a $N\times n$ random matrix $Y_n=(Y_{ij}^{n})$ where the entries are given by $Y_{ij}^{n}=\frac{\sigma(i/N,j/n)}{\sqrt{n}} X_{ij}^{n}$, the $X_{ij}^{n}$ being centered i.i.d. and $\sigma:[0,1]^2 \to (0,\infty)$ being a continuous…